Quick Notes

Quick Notes

Mathematics Chapter 9: Basic Polygons
9.3 Properties of Quadrilaterals and the Interior and Exterior Angles of Quadrilaterals
 
9.3  Properties of Quadrilaterals and the Interior and Exterior Angles of Quadrilaterals
 
The properties of a quadrilateral:
 

(i) Rectangle

  • The number of axes of symmetry is \(2\).
  • The opposite sides are parallel and of equal length.
  • All of its interior angles are \(90^\circ\).
  • The diagonals are of equal length and are bisectors of each other.
 

 

(ii) Square

  • The number of axes of symmetry is \(4\).
  • All the sides are of equal length.
  • The opposite sides are parallel.
  • All of its interior angles are \(90^\circ\).
  • The diagonals are of equal length and are perpendicular bisectors of each other.
 

 

(iii) Parallelogram

  • The number of axes of symmetry is \(0\).
  • The opposite sides are parallel and of equal length.
  • The opposite angles are equal.
  • The diagonals are bisectors of each other.
 

 

(iv) Rhombus

  • The number of axes of symmetry is \(2\).
  • All the sides are of equal length.
  • The opposite sides are parallel.
  • The opposite angles are equal.
  • The diagonals are perpendicular bisectors of each other.
 

 

(v) Trapezium

  • The number of axes of symmetry is \(0\).
  • Only one pair of opposite sides is parallel.
 

 

(vi) Kite

  • The number of axes of symmetry is \(1\).
  • Two pairs of adjacent sides are of equal length.
  • One pair of opposite angles is equal.
  • One of the diagonals is the perpendicular bisector of the other.
  • One of the diagonals is the angle bisector of the angles at the vertices.
 

 
Determine the interior angles and the exterior angles of a quadrilateral:
 
  • The sum of the interior angles of a quadrilateral is \(360^\circ\).
  • The sum of an interior angle of a quadrilateral and its adjacent exterior angle is \(180^\circ\).
  • The opposite angles in a parallelogram (or rhombus) are equal.
 
Example

The following diagram shows that \(PQRS\) is a parallelogram and \(PST\) is a straight line.

Calculate the value of \(x\) and \(y\).

Noted that the opposite angles in a parallelogram are equal.

So, \(x=42^\circ\).

Also, the sum of an interior angle of a quadrilateral and its adjacent exterior angle is \(180^\circ\).

Thus,

\(\begin{aligned} x+y&=180^\circ \\\\42^\circ+y&=180^\circ \\\\y&=180^\circ-42^\circ \\\\&=138^\circ. \end{aligned}\)